`International Journal of Mathematics and Mathematical SciencesVolume 2008, Article ID 597408, 8 pageshttp://dx.doi.org/10.1155/2008/597408`
Research Article

Bipartite Toughness and ??-Factors in Bipartite Graphs

1School of Mathematics and System Sciences, Shandong University, Jinan 250100, China
2Department of Computer Science, Memorial University of Newfoundland, St. John's, NL, Canada A1B 3X5
3School of Computing, The University of Southern Mississippi, Hattiesburg, MS 39406, USA
4Department of Mathematics, College of Arts and Sciences, The State University of West Georgia, Carrollton, GA 30118, USA

Received 18 August 2008; Accepted 7 October 2008

Copyright © 2008 Guizhen Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define a new invariant in bipartite graphs that is analogous to the toughness and we give sufficient conditions in term of for the existence of -factors in bipartite graphs. We also show that these results are sharp.

1. Introduction

Toughness, like connectivity, is an important invariant in graphs. There has been extensive work on toughness (see the survey in [1]) since Chvátal introduced the concept in 1973 [2]. The toughness of a graph is the minimum value of , where is a proper subset of the vertices of and is the number of connected components after removing from . (If is a complete graph so that is always equal to 1, then is set to be .) That is, for any integer , cannot be split into connected components by removing less than vertices. We also say that is -tough. Chvátal made a number of conjectures in [2], including the famous 2-tough conjecture saying that every 2-tough graph has a Hamiltonian cycle. Having inspired many interesting results, the 2-tough conjecture itself was showed to be false by Bauer et al. in 2000 [3].

A subgraph of is called a factor of if is a spanning subgraph of . An important class of factors is -factors, also called regular degree factors, where every vertex of has degree in . (Note that a perfect matching is a 1-factor, and a Hamiltonian cycle is a connected 2-factor.) There has been extensive work on the conditions of existence of various factors in graphs. Many results can be found in the latest survey by Plummer [4].

It is natural to expect that toughness, yet another measure of the connectivity of a graph, ought to relate to the existence of -factors in graphs. Enomoto et al. [57] proved that every -tough graph contains a -factor if it satisfies trivial necessary conditions, and there are -tough graphs for any that do not contain a -factor. Consider a bipartite graph where is a partition of and is the edge set of with each edge having one end in and the other in . Katerinis [8] proved that every 1-tough bipartite graph has a 2-factor. Recall that the toughness of a bipartite graph is at most 1 because the removal of from (assuming ) results in an independent set . Therefore, it is not possible to use toughness to predict the existence of -factors in balanced bipartite graphs for any .

1.1. Bipartite Toughness

In this paper, we introduce bipartite toughness, which is analogous to the concept of toughness but reflects the bipartition of . The bipartite toughness of a bipartite graph is the minimum value of , where is a proper subset of or and is the number of connected components after removing from . We set for complete bipartite graphs, just like for complete graphs.

A bipartite graph can have a regular degree factor only if . Therefore, in the rest of the paper, we consider only a balanced bipartite graph with . For a subset of , we use to denote the set of vertices adjacent to at least one vertex in . For two disjoint subsets and of , we use to stand for the number of edges having one end in and the other in . Other terminologies and notations used in this paper follow [9] and other references.

Bipartite toughness measures the connectivity of a bipartite graph better than toughness does. In contrast to toughness that is at most 1 in a bipartite graph, can be arbitrarily big. For example, in a complete bipartite graph with one edge deleted, which approaches to , is just like in a complete graph with one edge deleted. Interestingly, a better invariant to predict the existence of -factors in balanced bipartite graphs, for any . Furthermore, by their definitions, calculating in a bipartite graph is easier than calculating since one is a subtask of the other.

1.2. Our Results

Let be a balanced bipartite graph with and be an integer. In this paper, we prove the following three theorems.

Theorem 1.1. Let . If , then has a 1-factor.

Theorem 1.2. For and , if , then has a -factor.

Theorem 1.3. For , if , then has a -factor.

These theorems together give a sharp bound of for to have a -factor, for . (See Figure 1. Note that when and is odd; and when .)

Figure 1: The bound of bipartite toughness in Theorems 1.2 and 1.3, illustrated with . The -axis is and -axis is . The bound is given by on the left and on the right of .

The bound of is sharp in the following senses.

(a)For Theorem 1.1, let and construct a balanced bipartite graph as follows. Let and , where , , and . Let be comprised of all possible edges between and and all possible edges between and . If is even, then we add into an edge between and . Here, so that by Lemma 2.1 below, has no 1-factor. On the other hand, it is not hard to verify that in this construction of . Therefore, is a sharp bound.(b)For Theorem 1.2, for integers and , construct a balanced bipartite graph as follows. Let and , where , , and . Let be comprised of all possible edges between and , all possible edges between and , and a 1-factor between and . Here, so that by Lemma 2.1 below, has no -factor. On the other hand, it is not hard to verify that in . Therefore, is a sharp bound.(c)For Theorem 1.3. Let and be an integer. Obviously, . Construct a balanced bipartite graph as follows. Let and where , , and . Let be comprised of all possible edges between and , all possible edges between and , and a -factor between and . Then . Again, by Lemma 2.1 below, has no -factor. Moreover, it is not hard to verify that . Therefore, is also a sharp bound.

It is also worth to mention that, unlike Enomoto et al.'s well-known result that -tough graphs have -factors, in our results the bound of is much smaller than , in fact less than 2 for most (see Figure 1). This looks counterintuitive but it is due to a (not so good) feature of . Although can approach to , most time it does not increase significantly with edge connectivity or minimum degree. For example, if , has minimum degree (say on vertex ), then removing all vertices in except would split into components. So even when is as high as .

2. Proofs of the Theorems

The following lemma will be needed in the proofs of theorems.

Lemma 2.1. Let be a balanced bipartite graph, where , and let be an integer. Then the following three statements are equivalent:
(i) has a -factor;(ii) has edge-disjoint 1-factors;(iii)for any and ,
Proof. (i) and (ii): following the König-Hall theorem [9, Theorem 5.2 and Lemma 5.2], a regular degree bipartite graph has a perfect matching. Therefore, a -factor of a bipartite graph can be partitioned into a collection of edge-disjoint perfect matchings (1-factors). (ii) to (i) is trivial.
(i) and (iii): the equivalence of (i) and (iii) can be deduced from the max-flow min-cut theorem [10, 11]. Convert into a network by (a) adding a source vertex with multiedges between and each vertex ; (b) adding a sink vertex with multiedges between and each vertex ; and (c) orienting each edge into a directed arc going from to , from to , or from to (see Figure 2). Clearly, has a -factor the network has a -flow from to any cut in the network that separates and contains at least forward edges. For any and , consider the cut shown in dashed line in Figure 2, we haveso that

Figure 2: For proof of Lemma 2.1, red dashed line is the minimum cut.

Proof. Suppose has no -factor and , we will infer that . According to Lemma 2.1, there exist and such that . Let and . ThenObviously, . We can further assume thatBecause, if , then we can let and and have , , and . By symmetry, this converts to the case of .
We then have two cases to consider. Case 1.
If , then by (2.3). By and (2.4), we have , where . ThusThis completes the proof of Theorem 1.1. (Note that when , we have only Case 1 to consider.)

Proof. Now suppose , by (2.5), we have . Let . Then by (2.3), . Let . Then and . Therefore,
Case 2. If , then we have . By (2.4) and (2.7),
Case 2. If , then we have . By (2.5) and (2.7),Case 2.
Let be the unique integer satisfyingBy (2.10), . By (2.3) and (2.11), there is a vertex that is adjacent to at most vertices in . Let so and . By (2.4) and (2.11), we have . Therefore,Define a function . It is easy to verify that, by the assumption of , . Since is a convex function, it follows that for . By (2.12),

This completes the proof of Theorem 1.2.

Proof. Indeed, we will prove that the result in Theorem 1.3 holds for all . The condition of in Theorem 1.3 is only because that is not as tight a bound as when .
Suppose has no -factor, we will infer that . According to Lemma 2.1, there exist and such thatwhere and . Like in the proof of Theorems 1.1 and 1.2, we can still assume (2.4).
Suppose is vertex in that is adjacent to the least number (denoted by ) of vertices in . By (2.14), we have . Then with (2.4), we further have . Let , then and . Therefore,
This completes the proof of Theorem 1.3.

3. Conclusion and Future Work

We have defined a new invariant in bipartite graphs called bipartite toughness and provided a sharp bound of it for a balanced bipartite graph to have a -factor, for from 1 through . We view this as a big improvement from using toughness to predict -factors in bipartite graphs, as toughness of a bipartite graph is at most 1 and it cannot predict -factors for any .

There is also research on computational complexity of toughness. In general, recognizing toughness of a graph is NP-hard [12]. Furthermore, 1-tough of graphs is also NP-hard [13], and even 1-tough of bipartite graphs is NP-hard [14] too. Toughness in claw-free (-free) graphs [15], 1-tough in split graphs [14], and toughness in split graphs [16] have been shown in P. In the future, it would be very interesting to determine the complexity of bipartite toughness.

Acknowledgment

The first author's work is partially supported by National Natural Science Foundation of China NSFC 10871119.

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